prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
In a party of $2020$ people, some pairs of people are friends. We define a person's popularity as the size of the largest group of people in the party containing them such that every pair of people in that group are friends. A person has a popularity number of $1$ if they have no friends. What is the largest possible n... | 1010 |
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\frac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$. | 446 |
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fas... | 793 |
Let $\triangle ABC$ be a triangle with points $D$, $M$, and $N$ on sides $BC$, $AB$, and $AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. Given that the area of $\triangle BMD$ is $9\, \text{cm}^2$ and the area of $\triangle DNC$ is $25\, \text{cm}^2$, compute the area of $\triang... | 15 |
Suppose that $x_1 + 1 = x_2 + 2 = x_3 + 3 = \cdots = x_{2008} + 2008 = x_1 + x_2 + x_3 + \cdots + x_{2008} + 2009$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \displaystyle\sum_{n=1}^{2008} x_n$. | 1005 |
Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \neq 0$. What is the four-digit number $\overline{abcd}$? | 9376 |
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence? | 14 |
Let $S$ be the set of ordered pairs $(x, y)$ of positive integers for which $x + y \le 20$. Evaluate \[ \sum_{(x, y) \in S} (-1)^{x+y}xy. \] | 715 |
Let \[ S = \sum_{i = 1}^{2012} i! \]. The tens and units digits of \( S \) (in decimal notation) are \( a \) and \( b \), respectively. Compute \( 10a + b \). | 13 |
What is the largest positive integer $n$ less than $10,000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$. | 4369 |
In the diagram, congruent rectangles $ABCD$ and $DEFG$ share a common vertex $D$. The sides $BC$ and $EF$ intersect at point $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$, find the area of the region $ABHED$. | 54 |
For which natural number $k$ does \( \frac{k^2}{1.001^k} \) attain its maximum value? | 2001 |
Find the number of ordered pairs \((a,b)\) such that \(a,b \in \{10,11,\ldots,29,30\}\) and \(\text{GCD}(a,b)+\text{LCM}(a,b)=a+b\). | 35 |
The sequence $a_0$, $a_1$, $a_2$, $\ldots$ satisfies the recurrence equation: \[ a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3} \] for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$? | 181 |
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$? | 4 |
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. | 2017 |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2 + y^2 = 200$ and \[ \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \] is an integer? | 12 |
Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both $70$-character strings, as follows: \[ r = \frac{\text{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol}} {\text{lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}}. \] If $o=... | 1 |
Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such th... | 10087 |
Compute the number of ordered pairs of non-negative integers \((x, y)\) which satisfy \(x^2 + y^2 = 32045.\) | 4 |
Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\triangle ABC$ out of paper and then folding $\triangle AFD$ up along $FD$, folding $\triangle BED$ up along $DE$, and folding $\triangle CEF$ up alon... | 80 |
Evaluate the integral: $$\int_0^5 \max(2x, x^2) \, dx$$ | 43 |
Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b} \, \underline{c} \, \underlin... | 64 |
Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying the following equations: \[ \begin{align*} &a + b - c - d &= 3, \\ &ab - 3bc + cd - 3da &= 4, \\ &3ab - bc + 3cd - da &= 5. \end{align*} \] Find the value of $11(a - c)^2 + 17(b - d)^2$. | 63 |
Find the smallest positive integer $n$ such that $n^2 + 4$ has at least four distinct prime factors. | 94 |
A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For every distinct pair of lights $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 < x_2$ and $y_1 > y_2$, all lights strictly between them (i.e., $x_1 < x < x_2$ and $y_2 < y < y_1$) are toggled. After this procedure... | 1022121 |
The **Collatz function** is defined as: $$C(n) = \begin{cases} 3n - 1 & \text{if } n \text{ is odd}, \\ \frac{n}{2} & \text{if } n \text{ is even}. \end{cases}$$ We obtain the **Collatz sequence** of a number by repeatedly applying the Collatz function to that number. For example, the Collatz sequence of $13$ begins wi... | 21 |
Let $ABC$ be a triangle, and let $M$ be the midpoint of side $AB$. If $AB$ is $17$ units long and $CM$ is $8$ units long, find the maximum possible value of the area of $\triangle ABC$. | 68 |
Let $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$. | 595 |
Alexander has chosen a natural number $N > 1$ and has written down, in increasing order, all its positive divisors $d_1 < d_2 < \ldots < d_s$ (where $d_1 = 1$ and $d_s = N$). For each pair of neighboring numbers, he has found their greatest common divisor. The sum of all these $s-1$ numbers (the greatest common divisor... | 3 |
Which positive integers satisfy that the sum of the number's last three digits added to the number itself yields $2029$? | 2019 |
Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle FPQ$. | 210 |
A circle is inscribed in the square $ABCD$, where each side measures $10$ cm. The circle intersects sides $BC$ and $AD$ at points $M$ and $N$, respectively. Let $I$ be the intersection point of line $AM$ with the circle, distinct from $M$. Point $P$ is the orthogonal projection of $I$ onto line $MN$. Determine the leng... | 4 |
You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(... | 1195 |
Consider the sequence of six real numbers: 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? Note: The median of a sequence of six real numbers is the average of the two middle numbers after all the nu... | 135 |
Let $x_1, x_2, x_3$ be the solutions to the equation $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of \[ \sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right]. \] | 12033 |
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2 + 4px - p + 1$ an integer? | 4 |
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number? | 21 |
In a certain country, there are $2023$ islands and $2022$ bridges. Each bridge connects two different islands, and any two islands share at most one bridge. It is possible to travel from any island to any other island using these bridges. If, among any three islands, there exists an island connected by bridges to each ... | 1011 |
There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats such that neither little boy sits next to either little girl. Determine the number of different ways these six people can be seated. | 176 |
Let two positive integers $x, y$ satisfy the condition $44 \mid (x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$. | 21296 |
Michael is jogging along the beach when he observes a group of girls playing a game around an ellipse drawn in the sand. The ellipse has two foci, represented by two softball bats that are $24$ feet apart. Two children stand at the points where the ellipse intersects the line containing the bats, and these points are $... | 64 |
A triangle $ABC$ is given in which $\angle BAC = 40^\circ$ and $\angle ABC = 20^\circ$. Find the length of the angle bisector drawn from the vertex $C$, given that the sides $AB$ and $BC$ differ by $4$ centimeters. | 4 |
The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns: 1. A digit that is repeated 3 or more times in a row. For example, $0001$ and ... | 9720 |
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$... | 31 |
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations? | 683 |
Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her f... | 45 |
Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, \ldots, r_{10}$, and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i} + \frac{1}{r_j} - \frac{1}{r_i r_j}$ where $i < j$ and $i, j \in \{1, \ldots, 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $\frac{a}{b}$ for r... | 19 |
The number of positive integer pairs \((a,b)\) such that \(a\) divides \(b\) and \(b\) divides \(2013^{2014}\) can be expressed as \(2013n+k\), where \(n\) and \(k\) are integers with \(0 \leq k < 2013\). What is \(k\)? Recall \(2013=3\cdot 11\cdot 61\). | 27 |
Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BPD$? | 60 |
Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$? | 200 |
Let $f(n)=\left|\binom{n}{0}+\binom{n}{3}+\dots +\binom{n}{3\left[\frac{n}{3}\right]}-\frac{2^n}{3}\right|$, where $[x]$ is the greatest integer not exceeding $x$. Find $f(1)+f(2)+\dots+f(2021)$. | 898 |
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose $$(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.$$ Compute the product of all possible values of $a$. | 2009 |
Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges. | 15 |
Calculate the single-digit result obtained from the repeated sum of the digits of the number $2^{2004}$. Follow these steps: 1. Calculate the sum $S$ of all digits of $2^{2004}$. 2. Calculate the sum of the digits of $S$ to obtain $R$. 3. Continue calculating the sum of the digits until a single-digit number is reached... | 1 |
Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always... | 16 |
The real numbers $x$, $y$, and $z$ satisfy the system of equations: \[ \begin{align*} x^2 + 27 &= -8y + 10z, \\ y^2 + 196 &= 18z + 13x, \\ z^2 + 119 &= -3x + 30y. \end{align*} \] Find $x + 3y + 5z$. | 108 |
Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon? | 640 |
Determine the number of angles $\theta$ between $0$ and $2\pi$, excluding integer multiples of $\pi/2$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order. | 4 |
For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate \(\sum_{k=1}^{2003} (a_k-b_k)\). | 40842 |
Let $a, b, c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a) = b + c$, $P(b) = c + a$, $P(c) = a + b$, and $P(a + b + c) = -16$, find $P(0)$. | 11 |
What are the last two digits of the decimal representation of $21^{2006}$? | 21 |
We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.) \[ \begin{array}{l} 1: \textbf{FUNCTION} \ f(A) \\ 2: \quad \textbf{FOR} \ i=1,\ldots, \operatorname{length}(A)-1: \\ 3: \quad\quad A[i]\leftarrow A[A[i]] \\ 4: \quad\... | 4 |
Let $\alpha$ denote $\cos^{-1}(\tfrac{2}{3})$. The recursive sequence $a_0, a_1, a_2, \ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$, $$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$ Suppose that the series $$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$ can be expressed uniquely as $\t... | 23 |
Initially, each cell in a $29 \times 29$ table contains the number $0$. A movement consists of selecting a $5 \times 5$ sub-table and adding $+1$ to every cell within this sub-table. Determine the maximum value of $n$ such that, after performing $1000$ movements, there exist four cells whose centers form the vertices o... | 120 |
Two knights are placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square. The knights are said to attack each other if the distance between the centers of the squares on which the knights are placed is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squar... | 1848 |
Let $N = \overline{abcd}$ be a positive integer with four digits. We define the **plátano power** of $N$, denoted as $p(N) = \overline{\alpha_1\alpha_2\ldots\alpha_k}$, to be the smallest positive integer that can be inserted between the numbers $\overline{ab}$ and $\overline{cd}$ such that the new number $\overline{ab... | 45 |
On a $4 \times 4$ board, the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andres chooses the biggest number from each row, and Pablo chooses the biggest number from each column. The same number can be chosen by both. Then, all chosen numbers are removed from the boar... | 78 |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator... | 49 |
How many subsets $A$ of $ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} $ have the property that no two elements of $A$ sum to $11$? | 243 |
How many integers between $2$ and $100$ have only odd numbers in their prime factorizations? | 49 |
A hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box? | 216 |
Say that an ordered triple $(a, b, c)$ is pleasing if: 1. $a$, $b$, and $c$ are in the set $\{ 1, 2, \dots, 17 \}$. 2. Both $b - a$ and $c - b$ are greater than $3$, and at least one of them is equal to $4$. How many pleasing triples are there? | 81 |
Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by an... | 4004 |
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken - w... | 31 |
Let $S$ be a set. We say $S$ is $D^\ast$-finite if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a witness of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have? | 120 |
If $\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$. | 2018 |
Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all t... | 87 |
For each integer from 1 to 2019, calculate the product of its digits. Compute the sum of all 2019 products obtained. | 184320 |
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m - 3n)^2 = 2019$. Find $3m + n$. | 46 |
For positive integers $1 \le n \le 100$, let \[ f(n) = \sum_{i=1}^{100} i \left\lvert i-n \right\rvert. \] Compute $f(54) - f(55)$. | 2080 |
The school store is running out of supplies, but it still has five items: one pencil (costing $1), one pen (costing $1), one folder (costing $2), one pack of paper (costing $3), and one binder (costing $4). If you have $10, in how many ways can you spend your money? (You don't have to spend all of your money, or any of... | 31 |
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number. | 8093 |
The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties: - $AB = 12$ - $BC = 50$ - $CD = 38$ - $AD = 100$ - $BE = 30$ - $CE = 40$ Find the length of the segment $ED$. | 74 |
If for the real numbers $x, y, z, k$ the following conditions are valid: $x \ne y \ne z \ne x$ and \[ x^3 + y^3 + k(x^2 + y^2) = y^3 + z^3 + k(y^2 + z^2) = z^3 + x^3 + k(z^2 + x^2) = 2008, \] find the product $xyz$. | 1004 |
Suppose $a$, $b$, and $c$ are real numbers that satisfy the equations: $$a + b + c = 5$$ and $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}.$$ Find the greatest possible value of $a^3 + b^3 + c^3$. | 125 |
Nine points form a grid of size $3 \times 3$. How many triangles can be formed with $3$ vertices at these points? | 76 |
In an $n \times m$ grid, 1 row and 1 column are colored blue, and the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n, m)$? | 96 |
In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$. | 84 |
Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR. ($A \oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2 = 12$ and $6 \oplus 7 = 110_2 \oplus 111_... | 1 |
Given that $20N^2$ is a divisor of $11!$, what is the greatest possible integer value of $N$? | 72 |
Let $ABC$ be a right-angled triangle with $\angle B = 90^{\circ}$. The length of the altitude $BD$ is equal to $12$. What is the minimum possible length of $AC$, given that both $AC$ and the perimeter of triangle $ABC$ are integers? | 25 |
Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$. | 50 |
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\frac{\pi}{6}$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? The volume of a tetrahedr... | 320 |
If $f(x)$ is a monic quartic polynomial such that $f(-1) = -1$, $f(2) = -4$, $f(-3) = -9$, and $f(4) = -16$, find $f(1)$. | 23 |
In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle. [asy] size... | 180 |
For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\sum_{n=1}^{2022} f(n) \mod 1000$. | 242 |
In a chess tournament, each of two players has only one game played. After 2 rounds, 5 players left the tournament. At the end of the tournament, it was found that the total number of games played is 100. How many players were there at the start of the tournament? | 105 |
Let $y_0$ be chosen randomly from $\{0, 50\}$, let $y_1$ be chosen randomly from $\{40, 60, 80\}$, let $y_2$ be chosen randomly from $\{10, 40, 70, 80\}$, and let $y_3$ be chosen randomly from $\{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $P$ be the unique polynomial... | 107 |
Let $ABC$ be a triangle with sides $51$, $52$, and $53$. Let $\Omega$ denote the incircle of $\triangle ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1$, $r_2$, $r_3$ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed $r_1 + r_2 + r_3... | 15 |
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